Syllabus for the Written Exam | In PDF |
Sample exam in current format | In PDF |
August 31, September 2, 2016.
The Mathematics Ph.D. program at Rutgers includes two qualifying examinations, a written exam and an oral exam. The written exam is taken first and covers advanced calculus, elementary topology (metric spaces, compactness, and related topics), and the material of 501 (real analysis), 503 (complex analysis), and 551 (algebra). It is offered twice a year, near the beginning of each semester.
The syllabus represents a common core of material required of all Rutgers Ph.D.'s. In particular, the exam is designed with the goal that a pass on this exam shows a level of mathematical knowledge and ability appropriate for teaching the central undergraduate classes in mathematics.
Each student is required to take the exam by the beginning of the student's second year; the program director may allow a student who has entered with less preparation than the norm to take the exam a specified number of semesters later.
The exam consists of three separate component exams on different topics: algebra; real analysis and elementary point set topology; and complex analysis and advanced calculus. Students who do not pass the exam do not need to retake individual components which they have already passed.
Students who fail this exam may take it again during the semester following the one in which the exam was failed. Students who fail on the second attempt or who do not take the exams on schedule (as determined by the program director) will not be allowed to continue in the Ph.D. program.
"Free" attempt for entering students Students beginning graduate work at Rutgers may take the written qualifying exam at the beginning of their first semester in the program. If such a student fails the exam, this will not count as one of the two attempts that the student is normally allowed; the student will be allowed two additional attempts at the exam.
The exam is a six hour written exam broken up into three exams, each of which is two hours long. The three exams cover, respectively, algebra, real analysis and elementary point set topology, and complex analysis and advanced calculus. Each of these two hour exams consists of two parts. Part I has 3 problems, each of which is mandatory, and part II has 2 problems, of which the student is expected to do one. Each student is expected to submit solutions to all 3 problems in part I, and 1 out of the 2 problems in part II.
A complete solution to the January 2011 exam is posted here to serve as a model for students to learn how much justification and detail one should strive to provide --- these solutions are not worked out in a timed setting as the solutions to the real exams are, so can afford to contain more complete arguments; students can get full or close to full credits when their solutions contain all the key ingredients, with perhaps less detail than those contained here.
Please note that this exam, like all exams before Summer 2014, followed a different format. A sample exam in the current format, culled from problems of earlier exams, can be found here in pdf format.
Prior versions of the exam (Old format)